Planar non-formality of the little discs operad in characteristic two

Abstract

We show that the little discs operad D 2 is not formal over F 2 as a planar (or non-symmetric) operad. We compute explicitly the homological obstruction using as chain model the cells of the spineless cacti operad.

On the formality of the little disks operad in positive characteristic

“…In relation to non-formality, we should also mention the recent results [40] of Paolo Salvatore who showed that the planar (that is, non-symmetric) little 2-disks operad is not formal in characteristic 2. He proved this by an explicit computation of the first obstruction to formality.…”

A Künneth theorem for configuration spaces

“…We conclude the claim by noting that the ∞-categorical colimit coincides in the homotopy category with the homotopy colimit [23,Theorem 4.2.4.1], and the natural map We strongly emphasize that this hypothesis is not a conjecture; indeed, it is known to fail in some cases [28,31]. On the other hand, by [18], the hypothesis holds for any 𝑚 and 𝑛 as long as 𝐺 and 𝐻 are both contractible and 𝑅 is a field of characteristic zero -see also [21,24].…”

“…We strongly emphasize that this hypothesis is not a conjecture; indeed, it is known to fail in some cases [28, 31]. On the other hand, by [18], the hypothesis holds for any $$m$$ and $$n$$ as long as $$G$$ and $$H$$ are both contractible and $$R$$ is a field of characteristic zero — see also [21, 24].…”

Little discs operads, graph complexes and Grothendieck-Teichmüller groups

“…Nevertheless, we may wonder whether E n -operads are formal as non-symmetric operads, which is enough for the study of mapping spaces over an E 1 -operad. The case n > 2 of this question is still open, but Salvatore has proved in [51] that E 2 -operads are not formal as non-symmetric operads over F 2 .…”